RESEARCH

When children first learn about addition and subtraction, it is informal. They focus their learning on the meaning of the concept. Adding means ‘joining groups together’ and subtraction means ‘taking away from a group’. When adding, children will select 2 or more number cards, make the groups counting by ones, join the groups together, then count the number in the joined group by ones. When subtracting, children will select 2 number cards, ask themselves ‘which number could I make a group of that will give me enough to take away a group of the other number?’, make 1 group counting by ones, take away a group of the other number counting the number in that group they took away, then count the number in the group they have left. Once children understand addition and subtraction, they begin to record their understanding using number lines, initially counting by ones, then using place value to bridge to tens numbers. In a Year 1 classroom recently, we had children with all of these levels of understanding! The board contain all of the levels, built up during the 10 minutes of Explicit Teaching with the children: Students then selected the level that challenged them, then use playing card to generate numbers to investigate adding and subtracting. Walking around the class, stopping to observe and question children, I noticed that some children had chosen the ‘black’ level, but were finding it very challenging. These children had demonstrated their understanding of adding and subtracting counting by ones on a number line, but were not yet ready to add and subtract using place value to bridge 10. What to do? There was obviously a step in between that the children did not have deep enough understanding of, that was holding them back. Of course, the step involved Place Value. We had been investigating Friends of 10, Partitioning and Place Value of Teen Numbers, and I knew these were the concepts that the children now needed to apply to addition and subtraction. I asked one child, who was adding 8 and 5, if he knew 8’s Friend of 10. He hesitated, thought hard, then said, ‘5?’ Aha! I knew that until he was more fluent in his Friends of 10, he would not be able to add and subtract bridging 10. ‘Ok,’ I said to the child, ‘You’re not going to add today. You’re going to investigate Friends of 10, then you’ll be able to add again.’ I gave the child a 10 frame and some counters. Because we had previously investigated Friends of 10, he quickly learnt to select a number, place the corresponding number of counters on the 10 frame, and record the number and its Friend of 10. The child remained sitting where he was, amongst other children who were still investigating adding and subtracting at just beyond their current level of understanding. Because every child was investigating at just beyond their current level of understanding, they were all investigating independently (took time to...

When people began needing to count items, they used sticks or stones with one-to-one correspondence. For example, they would collect 1 stick for every sheep in their flock. When they wanted to see if they still had all of their sheep, they would ‘count’, matching one stick for one sheep. If at the end of the ‘count’ they had sticks left over, they had lost some sheep. If at the end of the ‘count’ they had sheep left over, they had gained some sheep. Over time, people realised that the amount of sheep was the same as the amount of sticks, and so names and symbols could be assigned to each amount, and numbers were born. This is a fundamental and important concept in counting – that a number always represents the same amount. The earliest number systems were additive – Roman Numerals are one example. Roman Numerals are additive because we need to add or subtract values of the symbols to get the value of the number. XCVIII is 100 minus 10 (90) plus 5 plus 1 plus 1 plus 1 (98). Additive systems worked while people only needed to represent numbers of small values. By about 2000 years ago, civilisations were beginning to travel long distances by sea, and to study the skies. This meant they needed a more efficient way to represent large numbers. Multiplicative number systems were the solution because multiplication makes numbers large much faster than addition (because multiplication is not just a quick way to add!!). Multiplication by a number larger than 1 makes it a number of times larger. For example, if you multiplied me by 2, don’t imagine there are 2 of me, imagine I am 2 times larger. And dividing me by 2, don’t imagine I have been cut into 2 parts (in half) but that I am 2 times smaller or half the size. Which way would you have more money – by repeatedly adding $10 ($10, $20, $30), or repeatedly multiplying by 10 ($10, $100, $1000)? Ancient civilisations understood multiplication, and many created multiplicative place value systems. The Hindus (who were so-named because they lived in the Indus Valley) knew this, so they based their number system on multiplication. And what number did they decide to multiply by? They decided to multiply by 10, simply because they had 10 fingers. Other civilisations multiplied by 20 (fingers and toes) and by 5 (fingers on one hand). But why is the base 10 number system the one that the whole world uses today? The base 10 number system soon spread to the Arabic part of the world – and thus came to be called the Hindu-Arabic number system. And there it stopped for about 700 years. During this 700 years, Indian and Arabic civilisations made great progress in both mathematics and science, made possible because they were using a multiplicative number system. While Europe, who were still using the additive number system involving Roman Numerals, were experiencing the Dark Ages, making...

COUNTING AND GROUPING Mathematics in the Early Grades: Counting & CardinalityEducation Development Center, Inc research on early numeracy and counting. One-to-One Correspondence: Foundation LevelVictorian Department of Education Learning and Teaching Resources on early counting concepts. Difficulties Children Face When Learning to CountAnn Gervasoni from Australian Catholic University’s research finding several common difficulties and issues related to learning to count. PLACE VALUE The Development of Structure in the Number SystemNoel Thomas from Charles Sturt University, Bathurst NSW’s cross-sectional study assessing children’s understanding of the number system. Multiplicative Patterns on a Place Value ChartNew York State Common Core Mathematics Curriculum conceptual exploration of the multiplicative patterns of the base ten system, including the relationship to metric measurement. Number Sense Series: A Sense of ‘ten’ and Place ValueJenni Way’s article on the importance of a strong ‘sense of ten’ as a foundation for both place value and mental calculations. What is a Ten Frame and why is it a useful tool?South African Numeracy Chair Project at Rhodes University’s article with research and activities using 10 frames for developing early number relationships. Standard and non-standard place value partitioningHilder Road State School’s examples of partitioning numbers using standard and non-standard place value. Children’s Understanding of Two-Digit Place ValueUniversity of Virginia’s research finding that the development of young children’s number sense and understanding of the base 10 system is essential for the acquisition of more complex number skills in later years Place Value (page 7) University of Notre Dame Australia’s research on page 7 explores the complexity of standard place value. Rethinking Concrete ManipulativesClements and McMillen’s NCTM-published research finding the effect of using materials in understanding mathematical concepts depends on the...

TOWN is a NSW whole-class intervention initiative which focuses on improving numeracy skills of students in the upper primary years identified as not performing at expected stage level. A Learning Place A Teaching Place provides whole class resources focused on develop relational understanding of maths concepts, including in students in the upper primary years identified as not performing at expected stage level. For similarities and differences between TOWN and A Learning Place A Teaching Place, TOWN vs...

TEN is a NSW intervention program designed to provide support for students experiencing substantial difficulty in learning numeracy in the early years. A Learning Place A Teaching Place is a K-6 Mathematics Resource to provide relational understanding and metalanguage for all students, including those experiencing substantial difficulty in learning numeracy in the early years. For similarities and differences between TEN and A Learning Place A Teaching Place, TEN vs...

Both the NSW Numeracy Continuum (also known as PLAN DATA) and the Concept Sequences at A Learning Place A Teaching Place are based on Learning Research. On the NSW Numeracy Continuum, Aspects 1 and 2 ( are both based on Learning Research from 1996. The remaining Aspects are based on Learning Research from between 1998 (Place Value and Multiplication and Division) and 2007 (Fractions, Pattern and Number Structure, Measurement). All Concept Sequences at A Learning Place A Teaching Place are based on Current Learning Research, and are updated as new Research becomes available. For links between the NSW Numeracy Continuum and the Concept Sequences at A Learning Place A Teaching Place, follow this link. ...

Janette Bobis, University of SydneyMental computation is now receiving greater emphasis in the curriculum and in classrooms across the world than ever before. Janette Bobis’ research demonstrates the importance of the empty number line. Empty Number Line, The empty number line_ Making children’s thinking visible The National Numeracy Review, commissioned by the Australian Government, includes recommendations across curriculum and pedagogy. National Numeracy Review Karen Fuson, Northwestern UniversityKaren Fuson’s research suggests children do not learn place value concepts, or multi-digit addition and subtraction, even children who calculate correctly show little understanding of the procedures they are using. Karen C. Fuson: Issues in Place-Value and Multi-digit Addition and Subtraction Learning and Teaching (Free login to jstor.org) New Zealand Ministry of Education identifies that effective teachers of numeracy demonstrate some distinctive characteristics, including high expectations, emphasise connections between different mathematical ideas, promote the selection and use of strategies that are efficient and effective and emphasise the development of mental skills, challenge students to think by explaining, listening, and problem solving, encourage purposeful discussion, and use systematic assessment and recording methods to monitor student progress and torecord their strategies to inform planning and teaching. NZ Ministry of Education Adding It Up details the processes by which students acquire mathematical proficiency with whole numbers, rational numbers, and integers, as well as beginning algebra, geometry, measurement, and probability and statistics from a research about teaching for mathematics proficiency, focusing on the interactions between teachers and students around educational materials and how teachers develop proficiency in teaching mathematics. Adding It Up_National Academy of...

Professional Learning and Development involves change. Any change can involve concern. The Concerns-Based Adoption Model of Change (CBAM) explains the concerns that individuals may have at each level of change. Not everyone experiences concern at every level of the change process. The change process starts at the bottom. The first 3 levels of the change process are unconcerned, informational and personal. If an individual has concerns at these levels, they will be asking questions or making statements using ‘it’, for example, ‘I don’t know what it is’, ‘how does it fit in with what I am already doing?’, ‘how does it work?’, ‘what research is it based on?. The intervention to ensure individuals make it through the ‘it’ levels, is information. The individuals need time to gather information about the change – ‘it’. The next 3 levels of the change process are management, consequence, collaboration. If an individual has concerns at these levels, they will be asking questions or making statements using ‘I’, for example, ‘how do I do it?’, ‘am I doing it right?’, ‘how do I know?’, ‘how do I fit it in?’. The intervention to ensure individuals make it through the ‘I’ levels, is collaboration. The individuals need time to meet with others engaged in the change process to share how they are doing. The final level of the change process is refocusing. If an individual has concerns at this level, they will be asking questions or making statements, for example, ‘am I doing it the most effective way?’, ‘what else is there?’, ‘how else could I use it?’. These individuals need time to experiment with making the most of the...

Dylan Wiliam, Embedded Formative Assessment, highlights the importance of formative assessment as a tool to improve teacher practice and ultimately improve student learning. Clarifying, sharing, and understanding learning intentions and criteria for success. When students understand the learning intentions and what success looks like in the activity they are engaged in. Dylan Wiliam identified that success criteria based on process, can constrain as well as enhance learning. Too careful planning of process can prevent students from finding truly insightful solutions that we have not envisaged. Sometimes it is a good idea not to tell students what they are expected to find during the lesson! Learning intentions and success criteria could be viewed as floors rather than ceilings! Testing students to see if they have learned only what we have taught them narrows their capacity to show what they have really learned. It is important that students know where they are going, and what counts as quality work, but there cannot be a simple formula for doing this. Teachers professional judgement is paramount! Engineering effective classroom discussions, activities, and learning tasks that elicit evidence of learning. Before we can begin to teach students, we need to find out what the students already understand through assessment of, as and for learning. Students’ everyday work should be deigned to provide evidence of understanding. The difference between our students will become of how we teach. Providing feedback that moves learning forward. Feedback should not look backward at where students have come from. Feedback should provide the next steps. Activating learners as instructional resources for one another. Students should be helping one another learn. Sharing strategies and understandings means that students are learning not just one way, but many ways. Diversity is an asset! Individualised learning is being taken over by group learning. Increasing the amount of talk, questioning and engagement, increases learning. Activating learners as owners of their own learning. Increasing the cognitive level of engagement makes students active learners. Sharing levels of understanding with students allow them to determine and explain what they already understand, and what they are ready to learn...